From LIF to QIF: Toward Differentiable Spiking Neurons for Scientific Machine Learning

TL;DR

This work addresses the challenge of training spiking neural networks for continuous regression and PDE solving by introducing Quadratic Integrate-and-Fire (QIF) neurons as differentiable alternatives to Leaky Integrate-and-Fire (LIF). By embedding QIF dynamics into MLPs, DeepONets, and PINNs and contrasting against LIF in both direct training and ANN-to-SNN conversion, the authors demonstrate that QIF enables smooth gradient propagation and reduces jagged, non-physical artifacts in predictions. Across function regression, operator learning, and PDE benchmarks, QIF-based models deliver smoother, more accurate results and robust optimization, highlighting the potential of differentiable spiking dynamics to bridge neuroscience-inspired computation with physics-informed and operator-learning frameworks. The findings suggest that QIF can facilitate more reliable, scalable, and energy-efficient spiking architectures suitable for SciML, with promising implications for neuromorphic hardware and large-scale, event-driven computation.

Abstract

Spiking neural networks (SNNs) offer biologically inspired computation but remain underexplored for continuous regression tasks in scientific machine learning. In this work, we introduce and systematically evaluate Quadratic Integrate-and-Fire (QIF) neurons as an alternative to the conventional Leaky Integrate-and-Fire (LIF) model in both directly trained SNNs and ANN-to-SNN conversion frameworks. The QIF neuron exhibits smooth and differentiable spiking dynamics, enabling gradient-based training and stable optimization within architectures such as multilayer perceptrons (MLPs), Deep Operator Networks (DeepONets), and Physics-Informed Neural Networks (PINNs). Across benchmarks on function approximation, operator learning, and partial differential equation (PDE) solving, QIF-based networks yield smoother, more accurate, and more stable predictions than their LIF counterparts, which suffer from discontinuous time-step responses and jagged activation surfaces. These results position the QIF neuron as a computational bridge between spiking and continuous-valued deep learning, advancing the integration of neuroscience-inspired dynamics into physics-informed and operator-learning frameworks.

Figures (9)

• Figure 1: Architectures of MLP, PINN, DeepONet and their corresponding spiking counterparts.
• Figure 2: Spike generation and membrane potential dynamics of LIF (red) and QIF (blue) neurons.
• Figure 3: Plot of phase $\phi$ with respect to time $t$. $\phi(t)$ is in $[0,\phi_\Theta)$. The phase changes linearly between spike arrivals and experiences an instantaneous jump when a spike arrives. Once the threshold $\phi_\Theta$ is reached, a spike is emitted and phase reset to circle 'starting' point at $\phi_\text{reset}=0.$
• Figure 4: Parabola results: (a): Loss history in training by QIF neuron compared with direct SNN training by LIF neuron. (b) & (c): Parabola ground truth and predictions using QIF and LIF neurons.
• Figure 5: Ricker wavelet results: Comparison of predictions from QIF training, LIF direct training and conversion. The first row is comparison of QIF with LIF direct training and the second row shows their errors. The third row is the prediction and errors of conversion with 32 simulation steps.